{"id":"https://openalex.org/W7202393622","doi":"https://doi.org/10.48550/arxiv.2608.11211","title":"A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph","display_name":"A Forced-Structure Reduction and Verifiable Bounds for Conway's 99-Graph","publication_year":2026,"publication_date":"2026-07-13","ids":{"openalex":"https://openalex.org/W7202393622","doi":"https://doi.org/10.48550/arxiv.2608.11211"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.11211","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.11211","pdf_url":"https://arxiv.org/pdf/2608.11211","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.11211","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062854587","display_name":"Aalok Thakkar","orcid":"https://orcid.org/0000-0002-3195-585X","observed_orcids":["https://orcid.org/0000-0002-3195-585X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thakkar, Aalok","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5064938440","display_name":"Simone Severini B","orcid":"https://orcid.org/0000-0001-7305-6759","observed_orcids":["https://orcid.org/0000-0001-7305-6759"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Severini, Simone","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.5737000107765198,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.5737000107765198,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.24009999632835388,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.08540000021457672,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/strongly-regular-graphs","display_name":"strongly regular graphs","score":0.996999979019165},{"id":"https://openalex.org/keywords/circulant-graphs","display_name":"circulant graphs","score":0.8119999766349792},{"id":"https://openalex.org/keywords/automorphisms","display_name":"automorphisms","score":0.7829999923706055},{"id":"https://openalex.org/keywords/tabu-search","display_name":"tabu search","score":0.949999988079071},{"id":"https://openalex.org/keywords/search-space-reduction","display_name":"search space reduction","score":0.5550000071525574}],"concepts":[{"id":"https://openalex.org/C85847156","wikidata":"https://www.wikidata.org/wiki/Q59015987","display_name":"Verifiable secret sharing","level":3,"score":0.7293000221252441},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6146000027656555},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.515999972820282},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5008000135421753},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4957999885082245},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4505000114440918},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.4205000102519989},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41760000586509705},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3560999929904938},{"id":"https://openalex.org/C16311509","wikidata":"https://www.wikidata.org/wiki/Q4148050","display_name":"Dependency graph","level":3,"score":0.34290000796318054},{"id":"https://openalex.org/C161677786","wikidata":"https://www.wikidata.org/wiki/Q2478475","display_name":"Neighbourhood (mathematics)","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.3043999969959259},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.30169999599456787},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.29910001158714294},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28760001063346863},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.2874999940395355}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.11211","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.11211","pdf_url":"https://arxiv.org/pdf/2608.11211","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.11211","doi":"https://doi.org/10.48550/arxiv.2608.11211","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.11211","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.11211","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.11211","pdf_url":"https://arxiv.org/pdf/2608.11211","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306227","display_name":"Arthritis National Research Foundation","ror":"https://ror.org/03abbex33"},{"id":"https://openalex.org/F4320320671","display_name":"National Research Foundation","ror":"https://ror.org/05s0g1g46"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7202393622.pdf","grobid_xml":"https://content.openalex.org/works/W7202393622.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4409735225"],"related_works":["https://openalex.org/W7156635552","https://openalex.org/W7221467747","https://openalex.org/W7223477508","https://openalex.org/W2739473679","https://openalex.org/W7221817206","https://openalex.org/W7221334088","https://openalex.org/W2500888309","https://openalex.org/W6958372471","https://openalex.org/W7204893712","https://openalex.org/W7034178233"],"abstract_inverted_index":{"Conway's":[0],"99-graph":[1],"problem":[2],"asks":[3],"whether":[4],"a":[5,33,54,98,133,157],"strongly":[6],"regular":[7],"graph":[8,57,84],"with":[9],"parameters":[10],"$\\mathrm{srg}(99,14,1,2)$":[11],"exists.":[12],"We":[13,61,75,105],"develop":[14],"two":[15],"complementary":[16],"lines":[17],"of":[18,91],"attack.":[19],"Fixing":[20],"one":[21],"vertex,":[22],"the":[23,44,49,72,92,108,123,169,175,181],"conditions":[24],"$\u03bb=1$":[25],"and":[26,36,43,67,95,155,179],"$\u03bc=2$":[27],"force":[28],"its":[29],"neighbourhood":[30,42],"to":[31,126],"be":[32],"perfect":[34],"matching":[35],"determine":[37],"every":[38],"edge":[39],"between":[40],"that":[41,81,159],"remaining":[45],"vertices.":[46,60],"For":[47],"$(99,14,1,2)$,":[48],"unresolved":[50],"part":[51],"is":[52],"therefore":[53],"constrained":[55],"$12$-regular":[56],"on":[58,85],"$84$":[59],"encode":[62],"this":[63],"reduction":[64],"in":[65,151,187],"CP-SAT":[66],"validate":[68],"it":[69],"by":[70,78],"recovering":[71],"unique":[73],"$\\mathrm{srg}(9,4,1,2)$.":[74],"also":[76],"prove":[77],"exhaustive":[79],"enumeration":[80],"no":[82],"circulant":[83],"$\\mathbb{Z}/99$":[86],"satisfies":[87],"more":[88],"than":[89],"$68.0\\%$":[90],"CAISc":[93],"constraints,":[94],"we":[96,121],"give":[97],"validated":[99],"orbit":[100],"formulation":[101],"for":[102],"prescribed":[103],"automorphisms.":[104],"then":[106],"study":[107],"partial-score":[109],"search":[110,114,137,177],"problem.":[111],"Fourteen":[112],"human-designed":[113],"configurations":[115],"reached":[116],"at":[117],"most":[118],"$69.43\\%$.":[119],"Separately,":[120],"supplied":[122],"scoring":[124],"function":[125],"an":[127],"evolutionary":[128],"program-search":[129],"system.":[130],"It":[131],"produced":[132],"degree-preserving":[134],"$4$-vertex-switch":[135],"tabu":[136],"whose":[138],"best":[139,182],"verified":[140,183],"artifact":[141],"scores":[142],"$70.73\\%$.":[143],"The":[144],"generated":[145],"move":[146],"differs":[147],"from":[148],"those":[149],"used":[150],"our":[152,188],"own":[153],"searches":[154],"crosses":[156],"plateau":[158],"was":[160],"stable":[161],"under":[162],"them.":[163],"These":[164],"results":[165],"do":[166],"not":[167],"resolve":[168],"existence":[170],"problem,":[171],"but":[172],"they":[173],"reduce":[174],"exact":[176],"space":[178],"improve":[180],"partial":[184],"construction":[185],"found":[186],"experiments.":[189]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-14T00:00:00"}
